General Continuity Proof Question

[kathrynag]

Ok, let's say I had 3x^{2}-2x+1
I know we have lx-2l<\delta
Also l(x-2)(3x+4)l<\epsilon
My problem with these types of questions is dealing with the l3x+4l. I just don't really know what to do.

 

[mathman]

I'm not sure what you are looking for. However x~2 means that 3x+4~10. To be more precise |3x+4 - 10| < 3(delta).

 

[adriank]

We can restrict δ > 0 to be as small as we want, so let's say δ ≤ 1. Then using |x - 2| < δ ≤ 1, show that |3x + 4| < 13, so that |(x - 2)(3x + 4)| < 13δ. In order for |(x - 2)(3x + 4)| to be less than ε, you can just pick any δ ≤ ε / 13; δ = max{1, ε / 13} does it.

 

[HallsofIvy]

It is generally a good idea to actually say what you mean! you say "these types of questions" without saying WHAT types of questions. Probably you mean "prove that 3x^{2- 2x+ 1 is continuous at x= 3". that's what adriank assumed an he gave a very nice explanation.}

 

[kathrynag]

Ok, I understand that. I just wasn't sure what to do when I got a quadratic because I understood these questions just fine otherwise.